Probabilistic and average linear widths of Sobolev space with Gaussian measure

  • Authors:

  • Fang Gensun;Ye Peixin

  • Affiliations:

  • Department of Mathematics, Beijing Normal University, Beijing 100875, China;Department of Mathematics, Beijing Normal University, Beijing 100875, China

  • Venue:
  • Journal of Complexity
  • Year:
  • 2003

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Abstract

We determine the exact order of the p-average linear n-widths λn(a) (W2r, µ, Lq)p, 1 ≤ q ∞, 0 p ∞, of the Sobolev space W2r equipped with a Gaussian measure µ in the Lq-norm.Moreover, we also calculate the probabilistic linear (n, δ)-widths and p-average linear n- widths of the finite-dimensional space Rm with the standard Gaussian measure in lqm, i.e., λn,δ(Rm, vm, lqm)~m1/q-1/2 √m + ln(1/δ), 1 ≤ q 2, m ≥ 2n, δ ∈ (0, 1/2], λn(a) (Rm, vm, lqm)p~m1/q, 1 ≤ q ∞, 0 p ∞, m ≥ 2n, δ ∈ (0, 1/2]. For the case of 2 ≤ q ≤ ∞, Maiorov and Wasilkowski have obtained the exact order of the probabilistic linear (n, δ)-widths λn,δ(Rm, vm, lqm), 2 ≤ q ≤ ∞, and p-average linear n-widths λn(a) (Rm, vm, lqm)1,q = ∞, p = 1.